Add up all of a number's proper divisors (its divisors excluding itself), and you get a new number. Add up that new number's proper divisors too... keep doing this, and some curious numbers eventually loop back to the very number they started from.
For example, 6's proper divisors are 1, 2, 3, and adding them up gives 1+2+3=6 — right back to itself! A number like this, where the sum of its proper divisors equals itself, is called a perfect number — a loop of length 1, in other words.
Some pairs of numbers point at each other. A's proper-divisor sum becomes B, and B's proper-divisor sum becomes A again. A pair like this is called amicable numbers, and when 5 or more numbers cycle around pointing at each other, it's called sociable numbers. Most numbers, by contrast, are deficient or abundant, and never form a loop like this at all.